Super operators

Li is the matrix representation of the lattice Liouvillian in the space of the basis operators, 1 is a unit (super)operator and Ci are projection vectors representing the operators of Eq. (32) in the same space. The... 

Recall that in his Theorems 3 and 4 Hans Kummer [3] defined a contraction operator, L, which maps a linear operator on A-space onto an operator on p-space and an expansion operator, E, which maps an operator on p-space onto an operator on A-space. Note that the contraction and expansion operators are super operators in the sense that they act not on spaces of wavefunctions but on linear spaces consisting of linear operators on wavefunction spaces. If the two-particle reduced Hamiltonian is defined as... 

The last line defines the mixed quantum-classical Liouville (super)operator C. 

It is the super operator for the time-dependent relaxation constants. Notice that... 

There are alternative notations for the transformation. Although the one used above to indicate a 90x pulse applied to nucleus I is clear, it is more common to use an operator (technically, a superoperator, which operates on an operator) to designate the transformation. As we would expect, a (super) operator I affects only I components, not S. Also, the pulse flip angle is often specified only if it is other than 90°. For example,... 

Table 11.2 summarizes the effect of various (super)operators, given at the top, on the individual spin operators at the left. The product operators for a two-spin system are included explicidy even though their transformations can readily be deduced from the effects of the superoperators on the individual components of the product operator. 

Here, H is the spin Hamiltoniam of the radical ion pair (RIP), R is the relaxation super operator, and K is the reaction operator. In H, the effect of the Zeeman interaction within the Ru -moiety is most efficient in pair spin state mixing due to the strong anisotropy of the g-... 

In order to express Eq. (50) in a more compact form, it is useful to introduce the super-operator formalism (Pickup and Goscinski, 1973 Goscinski and Lukman, 1970). The time-independent operators are construed as elements in a super-operator space with a binary product... 

Here is the sum of the Liouville super operator and the relaxation super operator R. K is the the self-exchange superoperator... 

The abstract formalism introduced in this chapter builds the fundament of the theory of extended two-particle Green s functions. Our approach is very general in order to allow for a unified treatment of the different species of extended Green s functions discussed in the main part of this paper. Since the discussed propagators can be applied to a wide variety of physical situations, the emphasis of this chapter lies on the unifying mathematical structure. The formalism is developed simultaneously for (projectile) particles of fermionic and bosonic character. We will define the general extended states which serve to define the primary or model space of the extended Green s functions. We also define the /.j-product under which the previously defined extended states fulfil peculiar orthonormality conditions. Finally we introduce a canonical extension of common Fock-space operators and super-operators to the space of the extended states. 

The specification super-operator is common in quantum chemical emd physical literature for linear mappings of Fock-space operators. It is very helpful to transfer this concept to the extended states A, B) and define the application of super-operators by the action on the operators A and B. We will see later how this definition helps for a compeict notation of iterated equations of motion and perturbation expansions. In certain cases, however, the action of a super-operator is fully equivalent to the action of an operator in the Hilbert space Y. The alternative concept of Y-space operators allows to introduce approximations by finite basis set representations of operators in a well-defined and lucid way. 

We will only use super-operators constructed in the following way ... 

We call U the super-operator associated with the operator U. The application of the super-operator U to the extended state A, B) (for arbitrary operators A and B) is defined as follows ... 

The symbols and are understood as operations that map a Fock-space operator onto the corresponding super-operator and Y-space operator,... 

A Y-space operator corresponding to the super-operator U generally exists only if ip) and y ) are eigenstates of the operator U and not for operators without this property. 

The same holds for the super-operator associated with a self-adjoint operator. 

So far we have used the picture of operators like H and p, acting on states like l rs) and Q/) in the Hilbert space Y. For developing perturbation theoretic expansions, however, it is useful to use the complementary concept of super-operators acting on Fock-space operators as defined in Sec. IIC. Using the super-operator H, the definition for the extended particle-hole Green s function of Eq. (23) can be written as... 

The matrix element of H that acts on the first and physical component of the extended states Yrs) indeed measures the excitation energy. The particular choice of the other components can best be motivated by the derivation via the super-operator H of Sec. IIC. The super-operator associated to the Hamiltonian operator H is denoted by H and acts on the extended states by giving the commutator of the Hamiltonian with the canonic operators al and... 

Under the condition that the reference state ip) is an eigenstate of the Hamiltonian H, the super-operator H can be replaced by the associated extended operator denoted by H and given by Eq. (51) ... 

In fact, the zeroth order extended states are eigenstates of the zeroth order excitation energy operator. Note that in contrast to the super-operator H, the perturbation expansion of the extended operator H contributes in all orders because the ground state energy Eo is a general function of the interaction strength. 

According to Eqs. (41) and (34) the static self energy (oo) is given by the primary block of the matrix of Eq. (33) minus its zeroth order part. An explicit expression can be found with the help of the super-operator formalism and using the definition of the extended states (47) ... 

Again, the contributions of the single-particle part of the Hamiltonian H may be split off easily due to the linearity H = Hq + v + V of the super-operators ... 

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